characteristics of the dynamics of mobile networks -- bionetics09

Overview MOSAR Project Dynamic Network Characterization Conclusion

Characteristics of the Dynamic of Mobile Networks

P. Borgnat, E. Fleury, J.-L. Guillaume, C. Robardet, A. Scherrerhttp://perso.ens-lyon.fr/eric.fleury/http://www.ens-lyon.fr/LIP/D-NET/

mailto://[email protected]

ENS Lyon/LIP – INRIA/D-NET

BIONETICSDecember 2009, Avignon, France

E. Fleury

http://perso.ens-lyon.fr/eric.fleury/

http://www.ens-lyon.fr/LIP/D-NET/

mailto://[email protected]


Outline

MOSAR ProjectProject overview

Dynamic Network CharacterizationMotivationStatistical analysis of snapshots of graphsTowards a global analysis of the dynamicsModeling of the dynamics

Conclusion

E. Fleury


Deployment of a large-scale dynamic networks

Control of antimicrobial resistance of bacteria responsible for majorand emerging nosocomial infections.

MOSAR ExperimentI Medical / staff / Patients (500 people)I Individual antibiotic use;I Characterization of the isolates

bacteria and their epidemicity;I 7/24 during 6 month long period

Document contact frequenciesI Associate 1 sensor with each actorI monitor the dynamic (inter & intra

contact)

E. Fleury


Patient room Patient room

E. Fleury


Multi modal / multi time scale

E. Fleury


Outline



Conclusion

E. Fleury


Objectives

MOSAR projectI Better understand the intrinsic characteristics / properties of

dynamic networksI Model / analyze interaction between node/usersI Describe accurately the dynamics

Two central questions:I Obtaining random models that

reproduce “these” propertiesI How do their functionalities constrain

the structures of real network?

E. Fleury


Preliminary data1

Mosar experiment: May 2009 – Nov 2009.

“Toy” traces are now availableI 41 nodes, 3 days (254 151 sec), every 120secI 820 possible links,I “[...] inter contact time distribution can be compared to the one of

power law [...]”

Power law...I What do power law really signify?I Is it the ultimate argument?

1A. Chaintreau and J. Crowcroft and C. Diot and R. Gass and P. Hui and J. Scott, Impact of Human Mobility on the Design of

Opportunistic Forwarding Algorithms, INFOCOM 2006

E. Fleury


Methodology

Descriptive: Standard graph properties

1 as a function of time to to provide an empirical statisticalcharacterization of the dynamics.

2 temporal evolution of the snapshots3 statistical signal processing

Analysis: global indicatorsI connected components, triangles, and communities

ModelI We propose models to perform random dynamic networks

simulations.

E. Fleury


Standard graph properties

Snapshots Gt = (V 0, Et)

I Active links: E(t) = |Et |I Connected vertcices: V (t) = |{u ∈ V 0, dGt (u) > 0}|I Average degree of connected vertices is D(t) =

∑u∈V 0 dGt (u)/V (t)

I Number of connected components (maximal subgraph such as everynode of the subgraph is connected to each another node): Nc(t) = |CGt |

I Number of triangles: T (t) = |TGt |

IMOTEProperty Mean Std. Dev. Corr.

Time (s)#Active links E(t) 21.9 12.4 5200#Connected vertices V (t) 19.9 4.7 7400Avg degree D(t) 2.1 0.8 3600#CC Nc(t) 4.8 2.1 5600#Triangles T (t) 6.9 8.30 4700

E. Fleury


Standard graph properties (cont)

Probability distributionI time bin of 1s << period.I PDF obtained are not heavy tailedI variability is not very large (stdv is a good measurement of the

variability)

0 1 2 3 4x 10

4

0

20

40

60

80

Time (seconds)

Edg

es

0 10 20 30 40 50 60 700

0.01

0.02

0.03

0.04

0.05

0.06

PD

F

Edges

E. Fleury



0 1 2 3 4x 10

4

0

10

20

30

40

Time (seconds)

Con

nect

ed v

ertic

es

0 5 10 15 20 25 30 350

0.02

0.04

0.06

0.08

0.1

0.12

PD

F

Connected vertices

Network is sparseI less than 10% of active links among the 820 possible linksI at no time the network is a single connected component.I many nodes remain isolated during long times (around 50% on

average for daytime and more than 90% for nighttime).

E. Fleury



0 1 2 3 4x 10

4

0

1

2

3

4

5

6

Time (seconds)

Ave

rage

Deg

ree

−1 0 1 2 3 4 50

0.2

0.4

0.6

0.8

PD

F

Average Degree

E. Fleury



differential sequence: DS[k ] = D[k + 1]− D[k ]

I log-log representation of the covariance in the wavelet domaina

I Sj is roughly the average of the wavelet coef. at scale jI Hurts exponent is close to the special value 0.5.I no long range −→ Independent Identically Distributed (IID)a

P. Abry and D. Veitch, Wavelet analysis of long-range dependent traffic, TIT, 1998

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

4

Diff

log 10

[P(D

iff=

x)]

1 2 3 4 5 6 7 8 9

−4

−2

0

2

4

6

8

j

log 2S

j

E. Fleury



0 1 2 3 4x 10

4

0

10

20

30

40

50

60

Time (seconds)

Num

ber

of tr

iang

les

0 10 20 30 40 500

0.05

0.1

0.15

0.2

PD

F

Number of triangles

Large number of triangles

I E(T (G(p, n))) =(N

3

)3!p3 & E(E(G(p, n))) = p N(N−1)

2

I When there is k links, E(T (G(n, k))) ∼ 8k3(N−2)N2(N−1)2

I 70 links (max) −→ 40(60)

I 22 links (avg) −→ 1(7)E. Fleury


Dynamical characteristics

Correlation timesI temporal evolution (X (t): univariate time-series)I The autocorrelation function of X (t):

CX (τ) =< X (t + τ)X (t) >t − (< X (t) >t)2

I correlation time: first time where the function CX (τ) goes tozero

notesI correlation times of E , V and Nc are rather large: ∼ 1h15.I D and T have comparable correlation times.I This suggests that these properties evolve under a common

cause.

E. Fleury


Dynamical characteristics (cont)

101

102

103

104

105

10−5

10−4

10−3

10−2

10−1

100

Contact Duration

P[X

>x]

101

102

103

104

105

106

10−5

10−4

10−3

10−2

10−1

100

Inter−contact Duration

P[X

>x]

Mean : 140; α = 1.66 Mean : 3680; α = 0.60

Contact and inter-contact durations

I P[X > x ] ∼x→∞

cx−α.

I α > 2: finite mean/variance; α < 2, infinite variance (heavy tailed).

I α < 1, infinite mean/variance.

E. Fleury


Dynamics of links creation and deletion

0 1 2 3 4x 10

4

0

2

4

6

8

Time (seconds)

Add

ed e

dges

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

log(

PD

F)

Added edges

I E⊕(t) = |{e ∈ Et , e /∈ Et−1}|, the number of links added at time t

E. Fleury


Dynamics of links creation and deletion (cont)

0 1 2 3 4x 10

4

0

2

4

6

8

10

Time (seconds)

Rem

oved

edg

es

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

log(

PD

F)

Removed edges

I E(t) = |{e ∈ Et−1, e /∈ Et}|, the number of links removed at time t

IMOTEProperty Mean Std. Dev. Corr. Time (s)

Edge creation E⊕(t) 0.15 0.55 680 ∼ 12minEdge delation E(t) 0.15 0.55 680∼ 12min

E. Fleury


Multivariate statistics of graph properties

Cross-correlationsI Strong influence E(t) over V (t);I Nc(t) related to E(t)I Less related: Nc(t) and V (t)I E⊕(t) and E(t): mostly uncorrelated

E(t) V (t) Nc(t) D(t) T (t) E⊕(t) E(t)E(t) 1 0.85 -0.56 0.95 0.90 0.19 0.15V (t) 0.85 1 -0.20 0.70 0.66 0.15 0.11Nc(t) -0.56 -0.20 1 -0.70 -0.41 -0.16 -0.15D(t) 0.95 0.69 -0.69 1 0.86 0.19 0.15T (t) 0.90 0.66 -0.41 0.86 1 0.15 0.11

E⊕(t) 0.19 0.15 -0.16 0.20 0.15 1 0.03E(t) 0.15 0.11 -0.15 0.16 0.10 0.03 1

E. Fleury



Joint distributionsI PXY (x , y) = P[X = x and Y = y ] = P[X = x/Y = y ]P[X = x ]

I variation of the # links is not constant over the # vertices

0 10 20 30 400

20

40

60

80

Number of connected vertices

Num

ber

of e

dges

0 10 20 30 400

10

20

30

40

50

60

70

80

Number of vertice

Num

ber

of e

dges

E. Fleury



Link correlationsI Most pairs of links have a very low correlation coefficient.

−0.4 −0.2 0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12x 10

4

Correlation coeficient

Num

ber

of e

dge

coup

les

Markovian evolution1 Correlation time link

creation/deletion is small2 Independent from the

evolution of other graphproperties

3 Links are independents

E. Fleury


Towards a global analysis of the dynamics

global propertiesI not directly interpretable in the sequence of static graphsI stability of connected componentsI communities embedded in the networkI proportion of creation of triangles

E. Fleury


Triangles in the graphs

P+/tri+ P+/tri= f+/tri+ f+/tri=IMOTE 44 % 56 % 6 % 94 %

RANDOM 10 % 90 % 5 % 95 %

links / trianglesI P+/tri+: link creation → triangleI f+/tri+: innactive link → triangleI 40% of link creations increase the number of trianglesI proportion of inactive links that would create a triangle is very lowI More potential links doesn not imply higher P+/tri+

E. Fleury


Modeling of the dynamics

Simulation algorithmI transition model with Markovian propertyI links e are independentI state of the networkI links e changes with Ptr (e, Gt)

I duration τ(e) since the link e has last changed its status

IngredientsI contact / inter contact duration distributionI elaborated graph properties (E(t), V (t), NC(t), D(t))I dynamical information (triangles)

E. Fleury


Modeling of the dynamics

Input: Simulation timeOutput: Random Dynamic Graphforeach Simulation Time Step t do

foreach link e doPtr (e, Gt) = TransitionProbability(e) given the state Gt ;pr = Uniform(0,1);if (pr ≤ Ptr (e)) then

ChangeState(e);end

endend

E. Fleury


Ingredients I

Contact distributionI heavy-tailed distributions for contact PON and inter-contact POFF

durationsI P+(τ): probability that one link that was OFF since τ (τ ≥ 1) is

activatedI PON(τ) = P−(τ)×

∏τ−1i=1 (1− P−(i))

P−(τ) =PON(τ)∏τ−1

i=1 (1− P−(i)), τ ≥ 2, P−(1) = PON(1) (1)

P+(τ) =POFF (t)∏τ−1

i=1 (1− P+(i)), τ ≥ 2, P+(1) = POFF (1) (2)

E. Fleury


Ingredients II

Imposed graph property distributionI Rejection Sampling based on a Metropolis-Hastings algorithmI new state G′

t = {Gt + Se(t) changed}, is accepted with probabilityPRS(Gt , G′

t) = min(

1,F (x(G′

t ))F (x(Gt ))

)I F is the target PDF for the graphI The total probability of transition of link e is then:

Ptr (e, Gt) = P−/+(τ(e)) · PRS(Gt , G′t).

E. Fleury


Ingredients III

Imposed dynamics of trianglesI reproduce the correct dynamical transition process concerning

trianglesI do not want to change the mean probabilities of transitionI The weighted probabilities are then:

Ptr (e, Gt) =

P+(τ(e))P+/tri=f+/tri=

for link creation without new triangle,

P+(τ(e))P+/tri+f+/tri+

for link creation with a new triangle.

E. Fleury


Simulation results

Investigated modelsI A: imposed empirical contact and inter-contact duration

distribution only.I B: imposed distributions of contact / inter-contact durations , and

of number of connected components.I C: distributions imposed contact / inter-contact durations and of

number of connected vertices.

0 2 4 6 8 10 120

50

Time (hours)

Num

ber

of e

dges

0 2 4 6 8 10 120

50

Time (hours)

Num

ber

of e

dges

102

103

10410

−4

10−3

10−2

10−1

100

Contact duration

P[X

>x]

103

104

10510

−4

10−3

10−2

10−1

100

Inter−contact duration

P[X

>x]

−− Imote / o Model A / ∗ Model B / + Model C

E. Fleury


Simulation results (cont)

0 20 40 60 800

0.02

0.04

0.06

0.08

0.1

0.12

Edges

PD

F

0 10 20 30 400

0.05

0.1

0.15

0.2

Connected verticesP

DF

0 2 4 6 8 10 120

0.05

0.1

0.15

0.2

0.25

Number of CC

PD

F

−− Imote / o Model A / ∗ Model B / + Model C

A: sole contact and inter-contact duration failsI the number of connected vertices is strongly over-estimatedI the number of connected components is under-estimated

E. Fleury



0 10 20 30 400

10

20

30

40

50

60

70

80

Number of vertice

Num

ber

of e

dges

0 10 20 30 400

10

20

30

40

50

60

70

80

Number of verticeN

umbe

r of

edg

es0 10 20 30 40

0

10

20

30

40

50

60

70

80

Number of vertice

Num

ber

of e

dges

A, B and C fail!I The density of the connected components (the groups) is

underestimatedI Links are spread uniformly in the graph

E. Fleury



0 10 20 30 400

10

20

30

40

50

60

70

80

Number of vertice

Num

ber

of e

dges

0 10 20 30 400

10

20

30

40

50

60

70

80

Number of verticeN

umbe

r of

edg

es0 10 20 30 40

0

10

20

30

40

50

60

70

80

Number of vertice

Num

ber

of e

dges

Weighted modelsI does not have an impact on the contact and inter-contact duration

distributionsI the density of connected components is comparable to the

experimental data

E. Fleury



0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

Density

PD

F

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

Density

PD

F

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

Density

PD

F

IMOTEA Bω

Density of frequent connected componentsI (τ = 7 and σ = 6)I classical models fail to create dense frequent connected

componentsI the number of frequent connected subgraphs is larger in the

simulated data than in the original

E. Fleury


Outline



Conclusion

E. Fleury


Conclusion

contributionsI rigorous / coherent set of properties (basic / advanced)I probability distribution of contacts and inter contacts is only one

parameterI global analyses to characterize the dynamics of the graph as a

whole:correlation between linksstability of the connected componentsnumber of trianglesevolution of communities inside the interaction networks.

I simple / accurate models that generate random interaction graphswith satisfactory temporal properties.

E. Fleury


Conclusion

Futur / On going worksI Introduce non-stationarity (piecewise stationary model)I Dynamic community computationI Overlapping community detectionI Trajectories of individuals as a signatureI Large in situ test beds to be deployed...

E. Fleury


Some references

Dynamic networks

I Antoine Scherrer, Pierre Borgnat, Éric Fleury, Jean-LoupGuillaume and Céline Robardet, Description and simulation ofdynamic mobility networks, in Computer Network 2008.

I Pierre Borgnat, Éric Fleury, Jean-Loup Guillaume, CélineRobardet and Antoine Scherrer, Analysis of Dynamic SensorNetworks: Power Law Then What?, in Comsware 2007.

E. Fleury

characteristics of the dynamics of mobile networks -- bionetics09

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